How does this continuity/mains tester work?

It's an LED turned on by a transistor! The base (gate?) lead connects to the screwdriver blade. Yes, all sorts of tricks and useful applications are possible with something this simple.

The resistor is a few-KVs high-volt type, probably chosen out of extreme safety paranoia (since customers might use it on a 440V breaker or worse. Can't have resistor arcing over internally!)

If the transistor is a JFET or an hfe~5000 Darlington, the ohms value can be extremely high: turn on 1mA LED using 50v signal input, Base resistor can be

50v/(1mA/5000) = 250meg

and 250v/250meg = 1uA, so it's pretty safe, electrocution-wise, plus offering lots of ESD protection to the transistor.

Try sweeping the driver blade across cloth or plastic sheet while touching the button. If the LED flickers, then the transistor is probably a JFET.

Traditionally such devices used a Neon bulb with a ~~~~ 100V strike voltage, a HIGH value resistor and capacitive body coupling to ground.

This one MAY instead use a MOSFET. AC capacitive couple signal turns on MOSFET by driving gate high enough and LED is driven via batteries. Gate sensitivity can be 1V or less if desired. If you can get enough voltage via capacitive coupling to strike a Neon then driving a MOSFET gate should be easy. And, apparently, is.

This should give you all you need to understand this useful little tool!